Canonical path integral measures for Holst and Plebanski gravity: I. Reduced phase space derivation

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Canonical path integral measures for Holst and Plebanski gravity: I. Reduced phase space derivation

Jonathan Engle, Muxin Han, Thomas Thiemann

To cite this version:

Jonathan Engle, Muxin Han, Thomas Thiemann. Canonical path integral measures for Holst and Ple-

banski gravity: I. Reduced phase space derivation. Classical and Quantum Gravity, IOP Publishing,

2010, 27 (24), pp.245014. �10.1088/0264-9381/27/24/245014�. �hal-00654153�

Canonical path integral measures for Holst and Plebanski gravity.

I. Reduced Phase Space Derivation

Jonathan Engle

, Muxin Han

, Thomas Thiemann

1

MPI f. Gravitationsphysik, Albert-Einstein-Institut, Am M ¨uhlenberg 1, 14476 Potsdam, Germany

2

Centre de Physique Th ´eorique

Campus de Luminy, Case 907, 13288 Marseille, France

3

Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, ON N2L 2Y5, Canada

4

Institut f. Theoretische Physik III, Universit ¨at Erlangen-N ¨urnberg Staudtstraße 7, 91058 Erlangen, Germany

Abstract

An important aspect in defining a path integral quantum theory is the determination of the correct measure. For interact- ing theories and theories with constraints, this is non-trivial, and is normally not the heuristic ”Lebesgue measure” usually used. There have been many determinations of a measure for gravity in the literature, but none for the Palatini or Holst formulations of gravity. Furthermore, the relations between different resulting measures for different formulations of gravity are usually not discussed.

In this paper we use the reduced phase technique in order to derive the path-integral measure for the Palatini and Holst formulation of gravity, which is different from the Lebesgue measure up to local measure factors which depend on the spacetime volume element and spatial volume element.

From this path integral for the Holst formulation of GR we can also give a new derivation of the Plebanski path integral and discover a discrepancy with the result due to Buffenoir, Henneaux, Noui and Roche (BHNR) whose origin we resolve.

This paper is the first in a series that aims at better understanding the relation between canonical LQG and the spin foam approach.

∗[email protected]

†[email protected]

‡[email protected], [email protected], [email protected]

§Unit´e Mixte de Recherche (UMR 6207) du CNRS et des Universit´es Aix-Marseille I, Aix-Marseille II, et du Sud Toulon-Var; laboratoire afili´e `a la FRUMAM (FR 2291)

Contents

1 Introduction 3

2 The path-integral measure for the Holst action 5

2.1 Reduced Phase space path integral . . . 5

2.2 Configuration path integral in terms of spacetimeso(η)-connection and tetrad . . . 7

2.2.1 Basic relations between variables . . . 7

2.2.2 Proof of bijectivity . . . 8

2.2.3 Rewriting the measure . . . 9

2.2.4 Final path integral . . . 10

3 The construction of path-integral measure for Plebanski-Holst, by way of Holst 11 3.1 Basic strategy and some definitions . . . 11

3.2 The coordinate transformation and its bijectivity . . . 12

3.3 The change of measure . . . 13

3.4 Final path integral . . . 15

3.5 An alternative way to construct Plebanski-Holst path-integral from Holst . . . 17

4 Consistency with Buffenoir, Henneaux, Noui and Roche 19

5 Discussion 22

A An example for checking the equivalences between the path-integrals of the Holst Hamiltonian, Ashtekar-Barbero- Immirzi Hamiltonian and triad-ADM Hamiltonian formalisms: Imposing the time-gauge 23

1 Introduction

Richard Feynman, in the course of his doctoral work, developed the path integral formulation of quantum mechanics as an alternative, space-time covariant description of quantum mechanics, which is nevertheless equivalent to the canonical approach [1]. It is thus not surprising that the path integral formulation has been of interest in the quantization of general relativity, a theory where space-time covariance plays a key role. However, once one departs from the regime of free, unconstrained systems, the equivalence of the path integral approach and canonical approach becomes more subtle than originally described by Feynman in [1]. In particular, in Feynman’s original argument, the integration measure for the configuration path integral is a formal Lebesgue measure; in the interacting case, however, in order to have equivalence with the canonical theory, one cannot use the naive Lebesgue measure in the path integral, but must use a measure derived from the Liouville measure on the phase space [2].

Such a measure has yet to be incorporated into spin-foam models, which can be thought of as a path-integral version of loop quantum gravity (LQG) [3, 4]. Loop quantum gravity is an attempt to make a mathematically rigorous quantization of general relativity that preserves background independence — for reviews, see [8, 6, 7] and for books see [9, 10]. Spin-foams intend to be a path integral formulation for loop quantum gravity, directly motivated from the ideas of Feynman appropriately adapted to reparametrization-invariant theories [4, 5]. Only the kinematical structure of LQG is used in motivating the spin-foam framework. The dynamics one tries to encode in the amplitude factors appearing in the path integral which is being replaced by a sum in a regularisation step which depends on a triangulation of the spacetime manifold. Eventually one has to take a weighted average over these (generalised) triangulations for which the proposal at present is to use methods from group field theory [3].

The current spin foam approach is independent from the dynamical theory of canonical LQG [11] because the dynamics of canonical LQG is rather complicated, it uses an apparently much simpler starting point: Namely, in the Plebanski formulation [14], GR can be considered as a constrained BF theory and treating the so called simplicity constraints as a perturbation of BF theory, one can make use of the powerful toolbox that come with topological QFT’s [12]. It is an unanswered question, however, and one of the most active research topics momentarily¹, how canonical LQG and spin foams fit together. It is one the aims of this paper to make a contribution towards answering this question.

In LQG one is compelled to introduce a 1-parameter quantization ambiguity — the so-called Immirzi parameter [15, 16].

This enters the action through a necessary extra ‘topological’ term added to the Palatini action; the full action is termed theHolst action [17]. To properly incorporate the Immirzi parameter into spin-foams, one should in fact not start from the usual Plebanski formulation but rather an analogous generalization, in which an analogous topological term is added to the action, leading to what we call the Plebanski-Holst formulation of gravity [19, 20, 21].

In [22] we have shown (and partly reviewed) for a rather general theory that different canonical quantisation techniques for gauge theories, specifically Dirac’s operator constraint method, the Master Constraint method and the reduced phase space method all lead to the same path integral. A prominent role in establishing this equivalence is played by what is called “the choice of gauge fixing” (from the reduced phase space point of view) or, equivalently, the choice of clocks (from the gauge invariant i.e.

relational point of view [24]). After a long analysis, it transpires that the common basis for the path integral measure no matter from which starting point it is derived, is the Liouville measure on the reduced phase, which can be defined via gauge fixing of the first class constraints. This measure can be extended to the full phase space and one shows that the dependence on the gauge fixing disappears when one integrates gauge invariant functions². From this point of view, that is, the equivalence between path-integral formulation and the canonical theory, it is obvious that formal path-integrals derived from the various formulations of gravity should all be equivalent, because all of them have the same reduced phase space — that of general relativity.

We thus apply the general reduced phase space framework to the Holst action as the starting point for deriving a formal path integral forboththe Holst action and the Plebanski-Holst action. It turns out that the resulting path-integral for either the Holst action or the Plebanski-Holst action is not naively the Lebesgue measure integral times the exponentiated action. There are extra measure factors of spacetime volume element Vand spatial volume elementVs. The presence of a spatial volume element is especially surprising because it breaks the manifest spacetime covariance of the path-integral when we are offshell.

The origin of this lack of covariance is in the mixture of dynamics and gauge invariance inherent to generally covariant systems

1Here we are referring the spin-foam model for 4-dimensional gravity, while for 3-dimensional gravity the consistency is discussed in e.g. [13].

2However, the dependence on the gauge fixing is secretly there,in a gauge invariant form, since choices of algebras of Dirac observables (i.e. gauge invariant functions) are in one to one correspondence with choices of gauge fixing. The choice of such an algebra is the zeroth step in a canonical quantisation scheme and determines everything else such as the representation theory, see [22] for a comprehensive discussion.

with propagating degrees of freedom and it is well known that the gauge symmetries generated by the constraints only coincide on shell with spacetime diffeomorphism invariance. The quantum theory chooses to preserve the gauge symmetries generated by the constraints rather than spacetime diffeomorphism invariance when we take quantum corrections into account (go offshell).

This kind of extra measure factor (so called local measure) has appeared and been discussed in the literature since 1960s (see for instance [25, 26]) in the formalism of geometrodynamics and itsbackground-dependentquantizations (stationary phase approximation). The outcome from the earlier investigations appears to be that in background-dependent, perturbative quantiza- tions, these measure factors ofVandVs only contribute to the divergent part of the higher loop-order amplitudes. Thus their meanings essentially depend on the regularization scheme used. One can of course try to choose certain regularization schemes such that, either the local measure factors never contribute to the transition amplitude, or that their effect is canceled by the divergence from the action [25, 26]. However, the power of renormalisation and the very reason we trust it is that its predictions are independent of the regularisation technique chosen. Therefore the status of these measure factors is very much unsettled, especially for non perturbative quantisation techniques. We here take the point of view that the measure factors should be taken seriously because they take the offshell symmetry generated by the constraints properly into account. In which sense this so called Bergmann – Komar “group” [33] is preserved in the path integral is the subject of the research conducted in [28]. In this article we confine ourselves to a brief discussion.

In the formalism of connection-dynamics, which is a preparation of background-independent quantization, a similar lo- cal measure factor also appears. It was first pointed out in [27], whose path-integral will be shown to be equivalent to our present formulation up to a discrepancy whose origin we resolve. When we performbackground-independentquantization as in spin-foam models, therefore the local measure factor should not be simply ignored, because the regularization arguments in background-dependent quantization have no obvious bearing in the background-independent context anymore. For example, spin-foam models are defined on a triangulation of the spacetime manifold with finite number of vertices, where at each vertex the value of local measure is finite, and the action also does not show any divergence.

However, so far none of the existing spin-foam models implements this non-trivial local measure factor in the quantization

3. The quantum effect implied by this measure factor has not been analyzed in the context of spin-foam models. But without it there is no chance to link spin foams with canonical LQG which at present is the only method we have in order to derive a path integral formulation of LQG from first principles. In ongoing work [32] we analyse the non-trivial effects caused by this measure factor in the context of spin-foam models, and try to give spin-foam amplitudes an unambiguous canonical interpretation by establishing a link between path-integral formulation and canonical quantization. In this article we also make a few comments on this.

The paper is organized as follows:

In section 2, after defining the reduced phase space path integral for a general theory, we begin with the Hamiltonian frame- work arising from theS O(η) Holst action⁴[36] (see also [37]). We then derive the path-integral formula for the Holst action in terms of spacetime field variables, i.e. theso(η) connectionω^{I J}_µ and the co-tetrade_µ^I.

In section 3, starting from the Holst phase space path integral, we construct a path-integral formula for the Plebanski-Holst action by adding some extra fields and extra constraints.

In section 4, we discuss the consistency with the calculations in [27].

Finally, we summarize and conclude with an outlook to future research.

3The ambiguities of the path integral measure in spin-foam models have been discussed in the literatures. In the context of spin-foam models, this issue of path integral measure can be translated into an ambiguity of the gluing amplitudes between 4-cells [29], while the quantum effects are discussed in [30]. And the relevance of the measure factor on the diffeomorphism symmetries of the spinfoam amplitudes is discussed in [31]. However in the present work we are concerning the measure factor which helps to make a connection with the canonical quantization, while the standard spin-foam approach doesn’t rely on the canonical framework and 3+1 splitting of the spacetime manifold.

4Our discussions apply to both Euclidean and Lorentzian signatures.

2 The path-integral measure for the Holst action

2.1 Reduced Phase space path integral

To cut a long story short (see e.g. [2, 22]) the central ingredient for most applications of the path integral is the generating functional

Z[j] :=

∫

DqDp|det({F, ξ})| √

det({S,S})δ[S]δ[F]δ[ξ] exp(i

∫

dt[p_aq˙^a+j_aq^a]) (2.1) Here (q^a,pa) denotes any instantaneous Darboux coordinates on phase space,S denotes the collection of all second class con- straintsS_Σ,Fthe collection of all first class constraintsF_µ,ξany choice of gauge fixing conditionsξ^µ, and jis a current which allows us to perform functional derivations at j =0 in order to define any object of physical interest. For instance the rigging kernel between initial and final kinematical statesψi(q), ψf(q) results by generating these two functions⁵ through functional derivation att =±∞. In addition, as usualDq =∏

t∈R,adq^a(t) andδ[F]=∏

t∈R,µδ(F_µ(t)), and likewise forDpandδ[S]. We will often write|D1|=[det(F, ξ)]², |D2|=det({S,S}). We will also drop the exponential of the current in what follows since it does not affect any of our manipulations, hence we will mostly deal with the partition functionZ= Z[0]. Since what one is really interested in isZ[j]/Zwe can drop overall constant factors from all subsequent formulas.

Applied to our situation, we restrict ourself to the case of pure gravity defined by the Holst action. We follow the notation employed in [17, 36]. Note that for the simplicity of the formulae, we skip “∏

x∈M” in almost all following path-integrals, where Mis the spacetime manifold. Moreover, we will assume that all the gauge fixing conditionsξαare functions independent of the connectionsω^{I J}a i.e. they are the functions of tetrad only. This assumption will simplify the following discussion. Then

Z=

∫

Dωa^{I J}Dπ^aI Jδ(C^ab)δ(D^ab) √

|D2|δ(G^{I J})δ(Ha)δ(H) √

|D1|∏

α

δ(ξ_α) expi

∫

dtd³x^(γ)π^aI Jω˙^{I J}a (2.2)

where^(γ)π^aI J :=(π−¹_γ ^⋆π)^a_{I J}, and the expressions of the constraintsG^{I J},Ha,H,C^ab, andD^abare given by [36]

GI J = Da

(γ)π^aI J:=∂a

(γ)π^aI J+ωaI^K

(γ)π^aJK−ωaJ^K

(γ)π^aIK

H_a = 1

2F_ab^{I J}[ω]^(γ)π^bI J

H = 1

4√

detq(F−1

γ ∗F)_ab^{I J}[ω]π^aIKπ^bJLη^KL C^ab = ϵ^{I JKL}π^a_{I J}π^b_KL

D^ab = 1 2√

detq ∗π^cI J(π^aIKDcπ^bJL+π^bIKDcπ^aJL)ηKL (2.3) whereD^ab is the secondary constraint with{H(x),C^ab(x^′)} = D^ab(x)δ(x,x^′). Note that the definition ofHandD^ab is slightly different from [36], up to a factor of 1/(2√

detq). In rewriting the kinematical Liouville measureDωa^{I J}D^(γ)π^aI J asDω^{I J}aDπ^a_{I J}, an overall constant Jacobian factor has also been dropped.

In Eq.(2.2)D2is the determinant of the Dirac matrix

 {C^ab(x),C^cd(x^′)} , {C^ab(x),D^cd(x^′)} {D^ab(x),C^cd(x^′)} , {D^ab(x),D^cd(x^′)}

=

 0 , {C^ab(x),D^cd(x^′)} {D^ab(x),C^cd(x^′)} , {D^ab(x),D^cd(x^′)}

 (2.4)

Therefore|D2|=[detG]²whereGis the matrix

G^ab,cd(x,x^′)={C^ab(x),D^cd(x^′)} ≈(detq)^3/2 [

q^abq^cd−1

2q^acq^bd−1 2q^cbq^ad

]

δ³(x,x^′). (2.5)

5Provided they are analytic. In case they are not, they are analytic functions times a reference vectorΩ0in which case the reference vector must be included in (2.1). See [22] for details.

By the symmetry ofq^ab, there exists an orthogonal matrix⁶M^absuch thatM^acM^bdq^cd=λ^aδ^abfor some{λ^a}, so that M^aeM^bfM^cgM^dhG^{e f,gh}=(detq)^3/2

[

λ^aλ^cδ^abδ^cd−1

2λ^aλ^bδ^acδ^bd−1

2λ^cλ^aδ^cbδ^ad ]

. (2.6)

Let ˆG^ab,cd :=λ^aλ^cδ^abδ^cd−¹₂λ^aλ^bδ^acδ^bd−¹₂λ^cλ^aδ^cbδ^addenote the portion in square brackets. Each of the rows (12),(13),(23) in Gˆhas exactly one non-zero matrix element. Reducing det ˆGby minors along these 3 rows,

det ˆG = Gˆ^12,12Gˆ^23,23Gˆ^13,13detR=−2⁻⁹(

λ¹λ²λ³)2

detR

whereR^ab=Gˆ^aa,bbis the reduced 3 by 3 matrix. detRhas only 2 non-zero terms; evaluating it and substituting in the result gives det ˆG= −1

4

(λ¹λ²λ³)4

= −1

4 (detq^ab)⁴= −1

4 (detq)⁻⁴, (2.7)

so that

detG= −1

4 (detq)^32×6(detq)⁻⁴ =−1

4 (detq)⁵. (2.8)

Thus, up to an overall factor, √

|D2|=(detq)⁵=:V_s¹⁰. (2.9)

Next we express the delta functionsδ(H) andδ(D^ab) in Eq.(2.2) as integrals of exponentials, Z =

∫

Dω^{I J}aDπ^aI JDNDdabV¹⁰_s δ(G^{I J})δ(Ha)δ(C^ab) √

|D1|∏

α

δ(ξα) expi

∫ dtd³x

[(γ)

π^aI Jω˙^{I J}a −NH+dabD^ab

].(2.10)

Then we follow the strategy used in [27] to eliminate the secondary second class constraintD^abin the path-integral. We consider a change of variables which is also a canonical transformation generated by the functional

F:=−

∫

d³x d_abC^ab/N. (2.11)

The integral measure is the Liouville measure on the phase space and thus is invariant under canonical transformation. √

|D2|, G_{I J},C^ab,√

|D₁|andξ_αare invariant because they strongly Poisson commute withC^ab(here we use the assumption that the gauge fixing conditions ξα only depend onπ^a_{I J}), andHa is invariant because it weakly Poisson commutes withC^ab. The change of kinetic termδ∫

dtd³x^(γ)π^aI J∂tω^{I J}a(x,t) is proportional to∫

dtd³x C^ab∂t(dab/N) which also vanishes by the delta functionsδ(C^ab) in front of the exponential. SoHandD^abare the only terms that change in the canonical transformation generated byF. Moreover because{H(x),C^ab(x^′)}=D^ab(x)δ(x,x^′) and{C^ab(x),D^cd(x^′)}=G^ab,cd(x,x^′) we can obtain explicitly the transformation behavior

6Here and later on in the paper, when we say a matrix is ‘orthogonal’, even if it has spatial-manifold indicies, we mean orthogonal in the standard matrix sense – i.e., ‘orthogonal’ with respect toδab, and not with respect to some covariantly determined metric.

ofH(N) andD^cd(dcd)

H(N)˜ ≡ ∑^∞

n=0

1

n!{F,H(N)}(n)

=

∫

d³x N(x)H(x)−

∫ d³x

∫

d³yN(x)

N(y)dab(y){C^ab(y),H(x)} + 1

2

∫ d³x

∫ d³y

∫

d³z N(x)

N(y)N(z)d_ab(y)d_cd(z){C^ab(y),{C^cd(z),H(x)}}

=

∫

d³x N(x)H(x)+

∫ d³x

∫

d³yN(x)

N(y)dab(y)D^ab(x)δ(x,y)

− 1 2

∫ d³x

∫ d³y

∫

d³z N(x)

N(y)N(z)dab(y)dcd(z)G^ab,cd(y,z)δ(x,z)

=

∫

d³x N(x)H(x)+

∫

d³x dab(x)D^ab(x)−1 2

∫ d³y

∫ d³z 1

N(y)dab(y)dcd(z)G^ab,cd(y,z) (2.12) D˜^cd(dcd) ≡ ∑^∞

n=0

1

n!{F,D^cd(dcd)}(n)

=

∫

d³x dcd(x)D^cd(x)−

∫ d³x

∫

d³y 1

N(y)dcd(x)dab(y){C^ab(y),D^cd(x)}

=

∫

d³x dcd(x)D^cd(x)−

∫ d³x

∫

d³y 1

N(y)dcd(x)dab(y)G^ab,cd(x,y) (2.13) here the series terminated because of{C^ab(x),G^{cd,e f}(x^′,x^′′)}=0. SinceG^ab,cd(x,y) is proportional toδ(x,y) we have

Z =

∫

Dω^{I J}a Dπ^a_{I J}DNDd_abV¹⁰_s δ(G^{I J})δ(H_a)δ(C^ab) √

|D₁|∏

α

δ(ξ_α) expi

∫ dtd³x

[

(γ)π^aI Jω˙a^{I J}−NH−1

2d_abd_cdG^ab,cd/N ]

=

∫

Dω^{I J}a Dω^{I J}t Dπ^aI JDN^aDNδ(C^ab) V_s¹⁰

√|det(G/N)|

√|D1|∏

α

δ(ξα) expi

∫ dtd³x

[(γ)

π^aI Jω˙^{I J}a −ω^{I J}t GI J−N^aHa−NH ]

=

∫

Dω^{I J}a Dω^{I J}t Dπ^aI JDN^aDNδ(C^ab)N³V⁵_s √

|D₁|∏

α

δ(ξ_α) expi

∫ dtd³x

[₍

γ)

π^aI Jω˙^{I J}a −ω^{I J}t G_{I J}−N^aH_a−NH

]. (2.14)

This is the canonical phase space path integral for the Holst action, with secondary constraints removed as in [38]. The Palatini case is recovered by settingγ=∞while holdingGconstant.

2.2 Configuration path integral in terms of spacetime so( η )-connection and tetrad

It is too difficult in concretely performing the integrations in Eq.(2.14) to compute transition amplitudes. However if we transform the Eq.(2.14) to be an integral of the Lagrangian Holst action in terms of original configuration variables, i.e. the spacetime connection fieldω^{I J}_µ and tetrad fielde^I_µ, the integral will become easier to handle. To rewrite the the canonical path integral as a configuration path-integral for the Holst action, one proceeds in two steps: (1.) Replace the canonical variables and Lagrange multipliers with space-time variables and the simplicity constraint (2.) Integrate out the simplicity constraint.

2.2.1 Basic relations between variables

In this section we give the definitions of the new coordinates in terms of the old coordinates. These definitions will be motivated and explained, and the bijectivity of the coordinate transformation demonstrated, in the subsequent section.

When the simplicity constraint is imposed,

C^ab=ϵ^{I JKL}π^aI Jπ^bKL≈0 (2.15)

π^a_{I J}takes one of the five forms⁷

(I±) π^aI J=±ϵ^abce_b^Ie_c^J (II±) π^aI J=±1

2ϵ^abce^K_be^L_cϵI JKL (2.16)

(Deg) π^aI J=0. (2.17)

Note that the appearance of the degenerated sector shows that the Hamiltonian constrained system derived from the Holst action is not regular, i.e. the rank of the Dirac matrix

 {C^ab(x),C^cd(x^′)} , {C^ab(x),D^cd(x^′)} {D^ab(x),C^cd(x^′)} , {D^ab(x),D^cd(x^′)}

 (2.18)

is not a constant on the whole phase space. We have to remove the degenerated sector in order to carry out the derivations in the reduced phase space. Therefore all the derivations in the last subsection hold only if the degenerate sector is removed. Now we restrict ourself in sector (II+), and in addition stipulate deteⁱ_a >0, removing the sign ambiguity in the definition ofe^I_a. The derivations for other sectors can be carried out in the same way. With the restriction to (II+), the above relation can be inverted as

e^I_a= 1 4√

2detπ^bo j⁻¹²ϵ^{I JKL}ϵabcπ^b0Jπ^cKL. (2.19)

This equation can then be used to definee^I_aoff-shell with respect to the simplicity constraint. One might ask whethere_a^I so defined, along withC^ab, form good coordinates onπ^a_{I J}. In fact, with the restrictions just stipulated, we will showπ^a_{I J} 7→(e^I_a,C^ab) is bijective in the next subsection.

Lastly, we equip the internal space with a time orientation, and definen^I as the unique internal future-pointing unit vector satisfyingn^Iπ^a_{I J}=0. Then one defines

e_t^I:=Nn^I+N^ae^I_a. (2.20)

When the simplicity constraint is satisfied,e^I_t is equal to thetcomponent of the physical space-time tetrad, so that the above definition is indeed an extension of the usuale^I_t.

2.2.2 Proof of bijectivity

For the purpose of making apparent the bijectivity of the coordinate transformation, and to aid in later calculations, define π^ai := 1

2π^a0i (2.21)

π˜^ai := 1

4ϵijkπ^ajk. (2.22)

In terms of these, the ‘triad’e^I_adefined in the last subsection can be alternatively introduced via 1. f_i^a:= detπ^bj⁻¹²π^ai, eⁱ_a=(f_i^a)⁻¹

2. e⁰_b:= 1

2ϵabcf_i^bπ˜^ci.

7To see that the four sectors (I±) and (II±) are disjoint, defineπ^a_i :=¹2π^a_0iand ˜π^a_i :=¹4ϵijkπ^a_jk. Then one has (I+) ⇒ detπ^ai =0 and (det ˜π^ai)(deteⁱ_a)>0, (I−) ⇒ detπ^ai =0 and (det ˜π^ai)(deteⁱ_a)<0, (II+) ⇒ det ˜π^ai =0 and (detπ^ai)(deteⁱ_a)>0, (II−) ⇒ det ˜π^ai =0 and (detπ^ai)(deteⁱ_a)<0.

Note that hereeⁱ_adenotes simply theI=1,2,3 components ofe_a^I,notthe co-triad.⁸

The mapf_i^a7→π^a_i is manifestly bijective. The definition ofe⁰_buses precisely the information contained in the anti-symmetric part of f_i^aπ˜^bi, whence the remaining information inπ^a_{I J}is exactly the symmetric part of f_i^aπ˜^bi:

S^ab := f_i^(aπ˜^b)i (2.23)

In terms of this, the simplicity constraint is given by

C^ab=−2π^(a_jπ˜^b)j=−2(dete^k_c)S^ab. (2.24) From this one sees thatπ^a_{I J} 7→(e^I_a,C^ab) is bijective.

2.2.3 Rewriting the measure We have

dπ^aI J=dπ^aid ˜π^ai (2.25)

The inverse of the relation betweenπ^a_i andeⁱ_a,π^a_i =¹₂ϵ^abcϵi jke_b^je^k_c, gives

∂π^a_i

∂e_b^j =ϵ^abcϵi jke^k_c. (2.26)

Note (ai) labels rows and (b j) labels columns. LetJ^aib jdenote this matrix. From thesingular value decomposition theorem, there exist orthogonal matricesO^abandOⁱjsuch thatO^baOⁱje_b^jis diagonal, that is

O^baOⁱje_b^j =λaδⁱa. (2.27)

LetO^aib j:=O^abO^ji. ThenO^aib jis also an orthogonal matrix, and we use it to define J˜^aib j:=O^aib jJ^{b j}ckO^ckdl =∑

c,k

ϵ^abcϵi jk(λcδ^kc)=∑

c

ϵ^abcϵi jcλc (2.28)

where the symmetry ofϵ^abcandϵi jkunder orthogonal transformations has been used. From the above equation, ˜J^ai_{b j} =0 when (i= j) or (a=b) or{i,j},{a,b}. From this one can deduce that, fori,a, the row (a,i) in ˜J^aib jhas only one non-zero element:

the one in column (b=i,j=a). Reducing by minors along these 6 rows then gives

det ˜J=J˜¹²21J˜²¹12J˜¹³31J˜³¹13J˜²³32J˜³²23detR (2.29) whereRⁱj=J˜ⁱⁱj jis the reduced 3×3 matrix. The diagonal elements ofRare zero, so that detRhas only 2 non-zero terms,

detR=J˜¹¹22J˜²²33J˜³³11+J˜¹¹33J˜²²11J˜³³22 (2.30) As one can check, ˜J¹¹22 = J˜²²11 =−J˜¹²12 = −J˜²¹21 =λ3, and similarly for cyclic permutations of 1,2,3. Plugging this into (2.30) and then (2.29) gives

det ˜J=2(λ1λ2λ3)³=2(deteⁱ_a)³. (2.31) detJ=det ˜J, so that dropping the irrelevant 2 factor,

Dπ^ai=(dete^k_c)³De_b^j. (2.32)

8One can see thateⁱ_amay not be taken as the co-triad from the following. Forv^a,w^btangent to the spatial sliceM, eⁱ_aebiv^aw^b=e^I_aebIv^aw^b+e⁰_aeb0v^aw^b=gabv^aw^b+σ²(e⁰_av^a)(e⁰_bw^b)≡qabv^aw^b+s(e⁰_av^a)(e⁰_bw^b) however,e⁰_ais in general arbitrary, so that the second term on the right hand side is in general non-zero, whence in general

eⁱ_aeai,qab.

Next, defineG^ab:= f_aⁱπ˜^bi, so that

∂G^ab

∂π˜^ci = f_i^aδ^bc. (2.33)

This is again block diagonal, whence

det (∂G^ab

∂π˜^ci )

=(detf_i^a)³ =(deteⁱ_a)⁻³ (2.34)

so that

Dπ˜^ai=(deteⁱ_a)³DG^ab =(deteⁱ_a)³DG^(ab)DG^[ab]=(deteⁱ_a)³DS^abDe⁰_b. (2.35) Lastly, from (2.24),DC^ab=(deteⁱ_a)⁶DS^ab, so that

Dπ˜^ai=(deteⁱ_a)⁻³DC^abDe⁰_b. (2.36) Coming to the lapse and shift, the Jacobian of the transformation (N,N^a)7→e^I_t is

J= ∂e^I_t

∂(N,N^a) =( n^I,e_a^I)

(2.37) On the other hand, the 4-volume element

dete_α^I =det( e^I_t,e^I_a)

=det(

NnI+N^aeaI,e^I_a)

=det( NnI,e^I_a)

=NdetJ (2.38)

thus|detJ|=V_sand dNdN^a=de^I_t/V_s.

Putting all the above relations together, we have

Dπ^aI JDNDN^a= 1

VsDe_µ^IDC^ab. (2.39)

Note the above measure isS O(η) covariant, consistent with theS O(η) covariance of the starting point.

2.2.4 Final path integral

Inserting (2.39) into (2.14), and integrating outC^abfinally gives Z =

∫

Dω_µ^{I J}De^I_µN³V_s⁴ √

|D1|∏

α

δ(ξα) expi

∫

e^I∧e^J∧ (

⋆FI J−1 γFI J

) [ω]

=

∫

Dω_µ^{I J}De^I_µV³Vs

√|D1|∏

α

δ(ξα) expi

∫

e^I∧e^J∧ (

⋆FI J−1 γFI J

)

[ω]. (2.40)

Note that the integral in Eq.(2.40) is restricted in the sector (II+). But if we want the integral to be over both the sectors (II+) and (II−), we will obtain

Z_± =

∫

II±Dω^{I J}_µDe^I_µV³V_s √

|D₁|∏

α

δ(ξ_α) cos

∫

e^I∧e^J∧ (

⋆F_{I J}−1 γF_{I J}

)

[ω]. (2.41)

In the existing spin-foam models in the literature [3, 18, 21], sectors (II+) and (II-) are not distinguished. One can see from the above equation, therefore, why it is generally the Cosine of the action and not the exponential of the action that is expected to appear (and does appear) in the asymptotic analysis of vertex amplitudes [39], see also the discussions of the issue in some other different perspectives [40]

In the follows, we always useZ±and∫

II±to denote the integral over both sectors, andZ,∫

only to denote the integral over a single sector (II+).

3 The construction of path-integral measure for Plebanski-Holst, by way of Holst

In this section we would like to relate the previous Holst action partition function with the partition function for the Plebanski- Holst action. Our starting point for the reconstruction is Eq.(2.14) (we first only consider a single sectorII+for simplicity)

Z=

∫

Dω^{I J}a Dω^{I J}t Dπ^a_{I J}DN^aDNδ(C^ab_ππ)N³V_s⁵ √

|D₁|∏

α

δ(ξ_α) expi

∫ dtd³x

[(γ)

π^aI Jω˙a^{I J}+ω^{I J}t G_{I J}−N^aH_a−NH ]

(3.1) where we use new notation for the simplicity constraintC^ab_ππ:=C^ab, anticipating the introduction of further simplicity constraints.

To remind the reader,

GI J = Da

(γ)π^aI J

Ha = 1

2F_ab^{I J}[ω]^(γ)π^bI J

H = 1

4√ detq

(F−1

γ∗F)^{I J}_ab[ω]π^a_IKπ^b_JLη^KL

C_ππ^ab = ϵ^{I JKL}π^aI Jπ^bKL. (3.2)

3.1 Basic strategy and some definitions

In order to rewrite this path integral as a (generalized) Plebanski path integral, one needs to change the variablesπ^aI J,N,N^ain favor of a constrained Plebanski two-formX^{I J}_µν. If we define≺Y,Z≻:= ¹₄ϵI JKLY^{I J}Z^KL, the constraint onX_µν^{I J}is

≺X_µν,X_ρσ≻= V

4!ϵ_µνρσ (3.3)

whereV:=ϵ^µνρσ≺X_µν,X_ρσ≻. This constraint impliesX_µν^{I J}takes one of the four forms X_µν^{I J} =

 ±2e^[I_µe_ν^J] (I±)

±ϵ^{I J}KLe_µ^Ke^L_ν (II±) (3.4)

for some tetrade_µ^I. On-shell,V=dete^I_µ, the 4-volume element. Following [27], we decomposeX^{I J}_µνinto π^aI J := 1

2ϵ^abc(Xbc)I J (3.5)

β^{I J}a := X^{I J}_ta (3.6)

and (3.3) becomes

C^ab_ππ := ≺π^a, π^b≻≈0

(C_ββ)ab := ≺βa, βb≻≈0 (3.7)

(C_βπ)^b_a := traceless part of ≺βa, π^b≻≈0.

The first of these constraints was imposed in section 2.2.1; the four sectors appearing there are the same four sectors here in (3.4).

As in section 2.2, we restrict to sector (II+). The last two of the constraints (3.7) are new.

Asπ^aI Jwas coordinatized bye_a^I and the simplicity constraintC_ππ^abin section 2.2, similarly in this section we introduce coordi- nates forβ^{I J}a. Specifically, we will define a change of variables

β^{I J}a ↔(N,N^a,C˜_ββ,C˜_βπ) (3.8)

where ˜C_ββand ˜C_βπhave the properties 1. IfC_ππ=0, then ˜C_βπ=C_βπ 2. If ˜C_βπ=0, then ˜C_ββ=C_ββ.